Optimal Partition of a Tree with Social Distance

TL;DR

This paper investigates the problem of partitioning trees to maximize social welfare based on social distance, providing a complete characterization for small-diameter trees and a linear-time solution, while establishing NP-hardness in 4-regular graphs.

Contribution

It offers a complete characterization of optimal partitions for small-diameter trees and presents a linear-time algorithm for trees, along with NP-hardness results for 4-regular graphs.

Findings

Optimal partitions characterized for small-diameter trees

Linear-time algorithm for MaxSWP on trees

NP-hardness of MaxSWP on 4-regular graphs

Abstract

We study the problem to find a partition of \textcolor{black}{a} graph $G$ with maximum social welfare based on social distance between vertices in $G$, called MaxSWP. This problem is known to be NP-hard in general. In this paper, we first give a complete characterization of optimal partitions of trees with small diameters. Then, by utilizing these results, we show that MaxSWP can be solved in linear time for trees. Moreover, we show that MaxSWP is NP-hard even for 4-regular graphs.